Compound Poisson approximation for triangular arrays with application to threshold estimation

Pavel Chigansky (The Hebrew University)
Fima Klebaner (Monash University)

Abstract


We prove  weak convergence of sums over triangular arrays to the compound Poisson limit using Tikhomirov's method.  The result is applied to statistical estimation of the threshold  parameter in autoregressive models.


Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-10

Publication Date: July 17, 2012

DOI: 10.1214/ECP.v17-2009

References

  • Alvo, Mayer; Cabilio, Paul; Feigin, Paul D. A class of martingales with nonsymmetric limit distributions. Z. Wahrsch. Verw. Gebiete 58 (1981), no. 1, 87--93. MR0635273
  • Arratia, Richard; Goldstein, Larry; Gordon, Louis. Poisson approximation and the Chen-Stein method. With comments and a rejoinder by the authors. Statist. Sci. 5 (1990), no. 4, 403--434. MR1092983
  • Barbour, A. D.; Chryssaphinou, O. Compound Poisson approximation: a user's guide. Ann. Appl. Probab. 11 (2001), no. 3, 964--1002. MR1865030
  • Berman, Simeon M. A compound Poisson limit for stationary sums, and sojourns of Gaussian processes. Ann. Probab. 8 (1980), no. 3, 511--538. MR0573291
  • Boutsikas, Michael V.; Vaggelatou, Eutichia. A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables. Bernoulli 16 (2010), no. 2, 301--330. MR2668903
  • Brown, B. M.; Eagleson, G. K. Martingale convergence to infinitely divisible laws with finite variances. Trans. Amer. Math. Soc. 162 1971 449--453. MR0288806
  • Chan, K. S. Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Ann. Statist. 21 (1993), no. 1, 520--533. MR1212191
  • Chan, Ngai Hang; Kutoyants, Yury A. Recent developments of threshold estimation for nonlinear time series. J. Japan Statist. Soc. 40 (2010), no. 2, 277--303. MR2830136
  • P. Chigansky and Y. A. Kutoyants, On nonlinear TAR processes and threshold estimation, Mathematical Methods of Statistics 21 (2012), no. 2, 142--152.
  • A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, International Statistical Review 70 (2002), no. 3, 419-435.
  • Hansen, Bruce E. Threshold autoregression in economics [discussion of MR2812802]. Stat. Interface 4 (2011), no. 2, 123--127. MR2812805
  • Ibragimov, I. A.; Hasʹminskiĭ, R. Z. Statistical estimation. Asymptotic theory. Translated from the Russian by Samuel Kotz. Applications of Mathematics, 16. Springer-Verlag, New York-Berlin, 1981. vii+403 pp. ISBN: 0-387-90523-5 MR0620321
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3 MR1943877
  • Kutoyants, Yury A. On identification of the threshold diffusion processes. Ann. Inst. Statist. Math. 64 (2012), no. 2, 383--413. MR2878912
  • Meyer, R. M. A Poisson-type limit theorem for mixing sequences of dependent "rare'' events. Ann. Probability 1 (1973), 480--483. MR0350816
  • Meyn, Sean; Tweedie, Richard L. Markov chains and stochastic stability. Second edition. With a prologue by Peter W. Glynn. Cambridge University Press, Cambridge, 2009. xxviii+594 pp. ISBN: 978-0-521-73182-9 MR2509253
  • Roos, Bero. Kerstan's method for compound Poisson approximation. Ann. Probab. 31 (2003), no. 4, 1754--1771. MR2016599
  • Serfozo, Richard F. Compound Poisson approximations for sums of random variables. Ann. Probab. 14 (1986), no. 4, 1391--1398. MR0866359
  • Serfozo, Richard F., Correction to: ''Compound Poisson approximations for sums of random variables'' [Ann. Probab. 14 (1986), no. 4, 1391--1398; MR0866359 (88d:60019)], Ann. Probab. 16 (1988), no. 1, 429--430. MR920283 (89a:60013)
  • Tihomirov, A. N. Convergence rate in the central limit theorem for weakly dependent random variables. (Russian) Teor. Veroyatnost. i Primenen. 25 (1980), no. 4, 800--818. MR0595140
  • Tong, Howell. Threshold models in nonlinear time series analysis. Lecture Notes in Statistics, 21. Springer-Verlag, New York, 1983. x+323 pp. ISBN: 0-387-90918-4 MR0717388
  • Tong, Howell. Threshold models in time series analysis—30 years on. Stat. Interface 4 (2011), no. 2, 107--118. MR2812802
  • Zolotarev, V. M. Estimates of the difference between distributions in the Lévy metric. (Russian) Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday, I. Trudy Mat. Inst. Steklov. 112 (1971), 224--231, 388. MR0321156
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.